Quantum Mechanics and the Particles of Nature: an outline for mathematicians by Anthony Sudbery has been one of my favorite books on quantum mechanics since I was an undergraduate. This is despite the fact that I probably haven't read it from cover-to-cover ever in my life. It has, however, been a book that I have returned to again and again when looking at quantum mechanics.
The book is a curious amalgam of physics and mathematics. The meat of the book, chapters 2-4, 6 and 7, are formally sound, mathematical beasts proceeding axiomatically though their various topics and become highly algebraic, in the modern sense. Chapter 3, on "Quantum Dynamics" introduces Lie algebras and symmetry groups (SU(2) and all that), and Sudbery never looks back. Most of the book proceeds with a pattern of: (1) description of the physics, (2) a theorem that encapsulates the description, and (3) a rigorous proof of the theorem. This is incredibly abstract for an introduction to the subject -- unless you're an undergraduate mathematics student, which this work was designed for. Sudbery is not trying to teach solution techniques, but rather to express how the artifice of quantum mechanics fits together logically, and this is often as much the province of applied mathematics as it is theoretical physics. I don't know the demarcation between the two, you'll have to get an applied mathematician and a theoretical physicist that are working on similar subjects to draw the line for you.
[It is designed for the British student, not the American one. I went through many undergraduate textbooks from England that I found extraordinarily good that were far too dense for any of my students. This of course may not be true for all colleges, but it was my experience.]
I have a few advantages over the usual reader for a book like this: (1) I survived my graduate Quantum Mechanics courses, (2) I have a degree in "theoretical mathematics" (as opposed to applied mathematics), so I am familiar with proofs and the axiomatic style and have even taken a modern algebra course, (3) I have taught both 300-level modern physics, materials science, and nanotechnology courses to undergraduates, all of which include a reasonable amount of quantum mechanics (even if very basic), and of course, (4) I talk about these sorts of things on a podcast. SO the going this time was easy, even if it is a book that I've owned for twenty-five years.
I originally found this book in the college bookstore -- we had an extraordinary college bookstore when I was an undergrad -- and immediately fell in love with it upon reading the first chapter. I'm sure I got through the next two and the fifth, but I don't know how much of the others. The first chapter, "Particle and Forces" is as lucid a description of particle physics as I've ever read, and it's a description that stayed with me even after I forgot where it came from. I found it so clear and memorable that I thought it came from a popular book, and when in the 2000s I was looking for it, I kept looking in old Scientific American books (Particles and Forces: at the heart of the matter, it think it was; it seems to have disappeared) and Polkinghorne's The Quantum World (which hasn't disappeared, and I'd replace if it did) and similar books that I'd read a little bit earlier or later. I did not go back to a textbook that I was working in my spare time and didn't actually finish. The reason why I wanted to find it again is that I'd attempted this book before I took a quantum mechanics course, and it left a lasting impression on how I thought about the subject. I finally found out that this was the book I had been looking for, off and on, for ten years when I pulled it out to help me prepare lectures for modern physics when I taught it.
But the thing that brought me back to this book most of all was chapter 5, "Quantum Metaphysics." This talks about the quantum theory of measurement, The de Broglie-Bohm Interpretation of Quantum Mechanics and Quantum Interpretations in a rigorous way. Even as an undergraduate, I found this chapter entrancing. When I took a philosophy course on (philosophical) cosmology, I used this general approach to discuss how the various interpretations of probability relate to the interpretations of quantum mechanics (which not think is a lot harder than I thought it was then). You can get an idea about how clear this chapter is if you listen to Randy and I talk about section 5.5 as the intermission in our discussion of Aharanov and Rohrlich's Quantum Paradoxes. It is a touchstone that I come back to every time I think about how to think about quantum mechanics.
That said, I had some rough going with the problems in this book. One reason is that so many of them are proofs, and a problem I've always had with proofs is determining whether or not I've really justified every step. Too often I worry that this or that step is for a special case and isn't extensible to other cases, and so on -- and I'm not as lucky as a student because I don't have a red pen man to tell me what I did wrong or reassure me that I did it right (students may disagree about the desirability of the red pen men). Another is that I think Sudbery expects you to have actually learned things in a mathematics course, meaning that I have to reconstruct the methods for the more application-oriented problems. This makes the problems especially valuable, and I find myself writing up multi-page analyses of what I did to solve a problem to remind myself what I did -- something I haven't done since my graduate work.
The chapters in between these two, 2-4, dealing with "Quantum Statics", "Quantum Dynamics", and "Some Quantum Systems" are all equally rewarding, giving a mathematical picture of the basic ideas of quantum mechanics. The last two chapters are more difficult. "Quantum Numbers" and "Quantum Field Theory" go a little beyond what is presented in most undergraduate courses. They talk about isospin and hypercharge, the weak and strong forces, and grand unification. All very interesting stuff that satisfy the "particles of nature" part of the title more than the "quantum mechanics. The integration of abstract algebra, matrix mechanics, and differential equations is complete here, and if you let your guard down reading about quantum metaphysics, it will bit you in the ass. It did me. I had a harder time with these two chapters than Cohen's An Introduction to Hilbert Space and Quantum Logic (available at Walmart , apparently), which I read a few years ago. I won't look it up to see how many years ago because I don't want to get depressed.
So again, if you're looking for a memorable book on quantum mechanics, one that you'll grow into, pick up Anthony Sudbery's Quantum Mechanics and the Particles of Nature. Even if it did cost me two weeks wages when I was working at the drug store, I never regretted picking this gem up.
Saturday, August 5, 2017
Tuesday, July 25, 2017
Sleeping Beauties in Theoretical Physics: 26 Surprising Insights by Thanu Padmanahhan:
Thanu Padmanahhan's book, Sleeping Beauties in Theoretical Physics is a great find. It's reasonably accessible for a technical tome on theoretical physics, it has an interesting framework that makes the connections between chapters on very different subjects coherent, and it covers a number of interesting topics, including ones that I'd thought I'd have references to in other books but my knowledge of them must have come from papers.
The feature that Padmanabhan uses to categorize physics is similar to the one used by Griffiths in his Introduction to Electrodynamics: a cube has eight vertices, and each of these vertices is a type of physical theory. Each of the eight vertices has one of three Boolean values: gravitation, relativity, and quantum. If the effect is accounted for, the value is on, if not, it's off. The vertex where they are all off is Newtonian Mechanics (without gravity) and the far vertex is a Theory of Everything (or somesuch, he calls it something else). Turning gravity off is like setting the gravitational constant equal to zero, turning relativity off is like setting the speed of light to infinity, and turning quantum mechanics off is like setting Planck's constant equal to zero.
So, I could make a list of the different vertices:
The most interesting statement in the introductory chapter is that GQM is a theoretically unpopular zone, and very poorly developed, despite being one step away from the TOE and possibly holding clues to it (or, I assume it would, even if it's boring to the high-powered mathematical physics wizards).
Sleeping Beauties touches on most of these topics (perhaps being short on the TOEs), showing up with some interesting takes on different phenomena, many of which I knew (rainbows and mirages, Thomas precession), many of which I didn't realize (the connection between Thomas precession and the Foucault pendulum), and many I had never touched on in my life (gravitational bending of the electric field). The twenty-six chapters have at least twenty-five topics and are each worthy of attention, especially those where he's making connections either between the vertices of his scheme or between subjects that seem disparate (but really aren't).
But, it's not for people who avoid mathematics.
The very first chapter of content, "The Emergence of Classical Physics," is an example of this. It tries to show how minimization principles in classical physics are explained by the limit of quantum mechanics. This uses Wigner functions to show that the action is the phase of the quantum mechanical wave function, and so classical particles follow trajectories that follow paths where the wave function's phases isn't cancelled out by neighboring paths (just like in Feynman's QED: The Strange Theory of Light and Matter). Obviously, if you're not up on the quantum mechanics or the calculus of variations, you might not see why this is so awesome. Speaking of QED, chapter 17, "If Quantum Mechanics is the Paraxial Optics, Then..." does something similar by applying the results of chapter 16's investigation of the transformation from wave optics to ray optics to discuss how quantum particles move in quantum field theory. Including why you must include trajectories for your particles that go backward in time. It's all the same process.
That is probably the most obvious recurring theme in the book: by understanding how a theory can be derived from a more fundamental theory, mysterious things like minimization principles can be made to make sense. That is, if you know how to turn off one of the switches, then you can better understand why things occur in the lower level theories -- and where those theories might lead you astray.
I therefore heartily recommend this book, and hope you both read and like it, but it does require an investment in mathematics to realize its physical returns.
["I therefore heartily recommend..." Looks like I've been writing too many recommendation letters, again.]
The feature that Padmanabhan uses to categorize physics is similar to the one used by Griffiths in his Introduction to Electrodynamics: a cube has eight vertices, and each of these vertices is a type of physical theory. Each of the eight vertices has one of three Boolean values: gravitation, relativity, and quantum. If the effect is accounted for, the value is on, if not, it's off. The vertex where they are all off is Newtonian Mechanics (without gravity) and the far vertex is a Theory of Everything (or somesuch, he calls it something else). Turning gravity off is like setting the gravitational constant equal to zero, turning relativity off is like setting the speed of light to infinity, and turning quantum mechanics off is like setting Planck's constant equal to zero.
So, I could make a list of the different vertices:
(0,0,0) Newtonian Mechanics
(1,0,0) Newtonian Gravity
(0,1,0) Special Relativity
(0,0,1) Qunatum Mechanics
(1,1,0) General Relativity
(1,0,1) Gravitational Quantum Mechanics
(0,1,1) Quantum Field Theory (1,1,1) Theory of Everything
The most interesting statement in the introductory chapter is that GQM is a theoretically unpopular zone, and very poorly developed, despite being one step away from the TOE and possibly holding clues to it (or, I assume it would, even if it's boring to the high-powered mathematical physics wizards).
Sleeping Beauties touches on most of these topics (perhaps being short on the TOEs), showing up with some interesting takes on different phenomena, many of which I knew (rainbows and mirages, Thomas precession), many of which I didn't realize (the connection between Thomas precession and the Foucault pendulum), and many I had never touched on in my life (gravitational bending of the electric field). The twenty-six chapters have at least twenty-five topics and are each worthy of attention, especially those where he's making connections either between the vertices of his scheme or between subjects that seem disparate (but really aren't).
But, it's not for people who avoid mathematics.
The very first chapter of content, "The Emergence of Classical Physics," is an example of this. It tries to show how minimization principles in classical physics are explained by the limit of quantum mechanics. This uses Wigner functions to show that the action is the phase of the quantum mechanical wave function, and so classical particles follow trajectories that follow paths where the wave function's phases isn't cancelled out by neighboring paths (just like in Feynman's QED: The Strange Theory of Light and Matter). Obviously, if you're not up on the quantum mechanics or the calculus of variations, you might not see why this is so awesome. Speaking of QED, chapter 17, "If Quantum Mechanics is the Paraxial Optics, Then..." does something similar by applying the results of chapter 16's investigation of the transformation from wave optics to ray optics to discuss how quantum particles move in quantum field theory. Including why you must include trajectories for your particles that go backward in time. It's all the same process.
That is probably the most obvious recurring theme in the book: by understanding how a theory can be derived from a more fundamental theory, mysterious things like minimization principles can be made to make sense. That is, if you know how to turn off one of the switches, then you can better understand why things occur in the lower level theories -- and where those theories might lead you astray.
I therefore heartily recommend this book, and hope you both read and like it, but it does require an investment in mathematics to realize its physical returns.
["I therefore heartily recommend..." Looks like I've been writing too many recommendation letters, again.]
Monday, July 17, 2017
Screwing Up on Quantum Computing
Well, I screwed that up.
In the next Physics Frontiers podcast, recorded about six months ago (approx. 1/27), Randy and I talked about photonics and quantum computing. The papers we read were very interesting and we had a good time talking about them, although we got a little bit confused (as my memory serves me, I believe that happens in the next episode as well). During the episode I went off on three digressions that I probably shouldn't have: science fiction, quantum computing, and economics and politics. I cut out all of the economics and politics bits (again). I left in the quantum computing, and that's what this is about.
I first read a journal article and had discussions with reasonably technical people about quantum computing in the 90's, before 1998 and after 1995 based on the people I was talking to. The whole idea that someone could construct an algorithm for a computer of a type that didn't exist yet really intrigued me at that point, and since one of my majors was "theoretical mathematics" (I think that's what the track name for all that abstract algebra and number theory stuff was at that school), figuring things about like how to factor large numbers very quickly was interesting to me. The applications to cryptography, not so much. But, the state of the art at that time was a sheet of paper and a proof, so my marginal interest in the subject didn't really go anywhere.
I've occasionally looked in on quantum computing from time to time since then. I've read a couple of technical books on the subject, for example, and I'll read articles about it when I actually have time to crack open one or two the Science and Nature magazines that clutter up my office. So, although I'm certainly not an expert (I'm not an expert about anything we talk about on Physics Frontiers), I certainly didn't think anything really astounding had happened. I was pretty sure a couple of people had put together a small number of qubits in one or the other of the ultracold settings.
I had no idea that Google and IBM had them up and running.
That's what my phone told me today, about lunchtime.
I think that I had heard that quantum computer simulators existed, and so some of my comments were just thoughtless ("people who design algorithms for computers that don't exist and then mathematically prove they'll run correctly" or something like that), but the fact that IBM has a 17 bit quantum processor up and running and they're giving people beta access with an SDK for Python makes some of my comments laughable. And of course, Google's up to the same thing.
It's amazing how fast these things are developing.
Oh well. The episode's already short, so I won't edit anything more out. You can just hear me babble from the past about things in the future.
Now, how can I get into that IBM beta?
In the next Physics Frontiers podcast, recorded about six months ago (approx. 1/27), Randy and I talked about photonics and quantum computing. The papers we read were very interesting and we had a good time talking about them, although we got a little bit confused (as my memory serves me, I believe that happens in the next episode as well). During the episode I went off on three digressions that I probably shouldn't have: science fiction, quantum computing, and economics and politics. I cut out all of the economics and politics bits (again). I left in the quantum computing, and that's what this is about.
I first read a journal article and had discussions with reasonably technical people about quantum computing in the 90's, before 1998 and after 1995 based on the people I was talking to. The whole idea that someone could construct an algorithm for a computer of a type that didn't exist yet really intrigued me at that point, and since one of my majors was "theoretical mathematics" (I think that's what the track name for all that abstract algebra and number theory stuff was at that school), figuring things about like how to factor large numbers very quickly was interesting to me. The applications to cryptography, not so much. But, the state of the art at that time was a sheet of paper and a proof, so my marginal interest in the subject didn't really go anywhere.
I've occasionally looked in on quantum computing from time to time since then. I've read a couple of technical books on the subject, for example, and I'll read articles about it when I actually have time to crack open one or two the Science and Nature magazines that clutter up my office. So, although I'm certainly not an expert (I'm not an expert about anything we talk about on Physics Frontiers), I certainly didn't think anything really astounding had happened. I was pretty sure a couple of people had put together a small number of qubits in one or the other of the ultracold settings.
I had no idea that Google and IBM had them up and running.
That's what my phone told me today, about lunchtime.
I think that I had heard that quantum computer simulators existed, and so some of my comments were just thoughtless ("people who design algorithms for computers that don't exist and then mathematically prove they'll run correctly" or something like that), but the fact that IBM has a 17 bit quantum processor up and running and they're giving people beta access with an SDK for Python makes some of my comments laughable. And of course, Google's up to the same thing.
It's amazing how fast these things are developing.
Oh well. The episode's already short, so I won't edit anything more out. You can just hear me babble from the past about things in the future.
Now, how can I get into that IBM beta?
Friday, June 30, 2017
List of Requirements for Gravitational Theories
Here it is without fanfare. If you have any questions about any part of it, please comment in the comments:
-------------------------------
To be theoretically consistent and compatible with experiment, a theory of gravitation must:
-------------------------------
To be theoretically consistent and compatible with experiment, a theory of gravitation must:
- Predict correct cosmological dynamics
- Big bang nucleosynthesis
- Produce the correct evolution of cosmological perturbations
- Cosmic microwave background
- Large scale structure
- Have the correct weak-field limits
- Reproduces Newtonian Mechanics
- Predicts post-Newtonian experiments in weak field
- Produces stable solutions
- Matter-side instabilities (Dolgov-Kawasaki)
- Ground states should be highly symmetric
- Gravity-side instabilities
- Stable de Sitter solutions
- Stability of the first loop in quantum gravity
- Stability in the face of inhomogeneous but isotropic perturbations
- Black hole nucleation
- Not contain any ghost fields
- Admit a well-posed Cauchy problem
- Reasonable theory of gravity waves
Monday, October 31, 2016
Physics Frontiers
Randy and I have started up a companion podcast, Physics Frontiers, with shorter discussions of topics in physics. Check it out! Show notes at the Physics Frontiers website.
Tuesday, August 4, 2015
Schrodinger's Cat
I've put together a video describing Schrodinger's cat and explaining why its important to have an interpretation of quantum mechanics. It doesn't give any answers, just explains why the question is so important. We overview some of the answers in the next podcast, which I will hopefully get up tomorrow (I'm putting in the breaks right now. I need to record the intro and upload it, so it's pretty likely).
Plus, I drew the pictures.
Plus, I drew the pictures.
Monday, July 6, 2015
Videos
I've got the first of the explanatory videos up. Randy and I have talked about putting up videos to help explain what is going on in the experiments for almost a year now. It can be really hard for someone to visualize what's happening in the experiments for the paradoxes just from our description. I've even found that in writing descriptions: the more precise someone describes something, the harder it is for someone else to understand it.
That's just not how we talk.
Unfortunately, the vaguer the description the more easily the listener misconstrues what is being said. And it turns out it's not an easy needle to thread: it takes many years for someone to be able to reliably read a geometrical description of a problem without getting confused.
But never fear, video is here! I can draw out what I want to talk about for you, as I have in this video.
That's just not how we talk.
Unfortunately, the vaguer the description the more easily the listener misconstrues what is being said. And it turns out it's not an easy needle to thread: it takes many years for someone to be able to reliably read a geometrical description of a problem without getting confused.
But never fear, video is here! I can draw out what I want to talk about for you, as I have in this video.
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